A map of the Solar System!

This is as close to canon as you can reasonably get, and was made from cross-referencing the various solar system models and orreries that appear in the show. Even if the order is wrong, I am 100% confident that there are five planets in the Solar System, plus one possible dwarf planet near the Sun. Things not represented: one of these planets has at least 14 moons. My bet would be Saturn (as it is canonically named), and the 'asteroids' are actually the planet's other moons. But I couldn't find anything to support this.

If we want to get technical: the small body orbiting the Sun is probably a dwarf planet, like Pluto. The first true planet seems to have purple oceans, and may have been depicted with an atmosphere in one of its appearances. Equus and her Moon need no description. The third planet is probably a greenhouse world, like Venus. Its apparent size on the models was far too small to be a gas giant. Planet four is Saturn, a ringed gas giant with at least one Moon confirmed by two different pieces of evidence. Planet five is a mystery. Its surface texture was like a paint splatter, and its relative size was bigger than Saturn. Its moon even appears to have an atmosphere. Lastly, the comet comes from Canterlot's orrery, and may represent the body responsible for the Centennial Meteor Shower as it makes its hundred-year journey around the system.